258 research outputs found
Elastic energy of polyhedral bilayer vesicles
In recent experiments [M. Dubois, B. Dem\'e, T. Gulik-Krzywicki, J.-C.
Dedieu, C. Vautrin, S. D\'esert, E. Perez, and T. Zemb, Nature (London) Vol.
411, 672 (2001)] the spontaneous formation of hollow bilayer vesicles with
polyhedral symmetry has been observed. On the basis of the experimental
phenomenology it was suggested [M. Dubois, V. Lizunov, A. Meister, T.
Gulik-Krzywicki, J. M. Verbavatz, E. Perez, J. Zimmerberg, and T. Zemb, Proc.
Natl. Acad. Sci. U.S.A. Vol. 101, 15082 (2004)] that the mechanism for the
formation of bilayer polyhedra is minimization of elastic bending energy.
Motivated by these experiments, we study the elastic bending energy of
polyhedral bilayer vesicles. In agreement with experiments, and provided that
excess amphiphiles exhibiting spontaneous curvature are present in sufficient
quantity, we find that polyhedral bilayer vesicles can indeed be energetically
favorable compared to spherical bilayer vesicles. Consistent with experimental
observations we also find that the bending energy associated with the vertices
of bilayer polyhedra can be locally reduced through the formation of pores.
However, the stabilization of polyhedral bilayer vesicles over spherical
bilayer vesicles relies crucially on molecular segregation of excess
amphiphiles along the ridges rather than the vertices of bilayer polyhedra.
Furthermore, our analysis implies that, contrary to what has been suggested on
the basis of experiments, the icosahedron does not minimize elastic bending
energy among arbitrary polyhedral shapes and sizes. Instead, we find that, for
large polyhedron sizes, the snub dodecahedron and the snub cube both have lower
total bending energies than the icosahedron
Some Exceptional Cases in Mathematics: Euler Characteristic, Division Algebras, Cross Vector Product and Fano Matroid
We review remarkable results in several mathematical scenarios, including
graph theory, division algebras, cross product formalism and matroid theory.
Specifically, we mention the following subjects: (1) the Euler relation in
graph theory, and its higher-dimensional generalization, (2) the dimensional
theorem for division algebras and in particular the Hurwitz theorem for normed
division algebras, (3) the vector cross product dimensional possibilities, (4)
some theorems for graphs and matroids. Our main goal is to motivate a possible
research work in these four topics, putting special interest in their possible
links.Comment: 14 pages, Late
IST Austria Thesis
The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the ÄŒech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's
A Prehistory of n-Categorical Physics
This paper traces the growing role of categories and n-categories in physics,
starting with groups and their role in relativity, and leading up to more
sophisticated concepts which manifest themselves in Feynman diagrams, spin
networks, string theory, loop quantum gravity, and topological quantum field
theory. Our chronology ends around 2000, with just a taste of later
developments such as open-closed topological string theory, the
categorification of quantum groups, Khovanov homology, and Lurie's work on the
classification of topological quantum field theories.Comment: 129 pages, 8 eps figure
Equicontinuous factors, proximality and Ellis semigroup for Delone sets
We discuss the application of various concepts from the theory of topological
dynamical systems to Delone sets and tilings. We consider in particular, the
maximal equicontinuous factor of a Delone dynamical system, the proximality
relation and the enveloping semigroup of such systems.Comment: 65 page
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